Candido Ferreira Xavier de Mendonça Neto

Splitting Number is NP-complete

We consider two graph invariants that are used as a measure of nonplanarity: the splitting number of a graph and the size of a maximum planar subgraph. The splitting number of a graph G is the smallest integer k ≥ 0, such that a planar graph can be obtained from G by k splitting operations. Such operation replaces a vertex v by two nonadjacent vertices v1 and v2, and attaches the neighbors of v either to v1 or to v2. We prove that the splitting number decision problem is NP-complete, even when restricted to cubic graphs. We obtain as a consequence that planar subgraph remains NP-complete when restricted to cubic graphs. Note that NP-completeness for cubic graphs also implies NP-completeness for graphs not containing a subdivision of K5 as a subgraph.

Vertex Splitting and Tension-Free Layout

In this paper we discuss the “vertex splitting” operation. We introduce a kind of “spring algorithm” which splits vertices to obtain better drawings. We relate some experience with the technique.

The Splitting Number of the 4-Cube

The splitting number of a graph G is the smallest integer k ≥ 0 such that a planar graph can be obtained from G by k splitting operations. Such operation replaces v by two nonadjacent vertices v1 and v2, and attaches the neighbors of v either to v1 or to v2. The n-cube has a distinguished plaice in Computer Science. Dean and Richter devoted an article to proving that the minimum number of crossings in an optimum drawing of the 4-cube is 8, but no results about splitting number of other nonplanar n-cubes are known. In this note we give a proof that the splitting number of the 4-cube is 4. In addition, we give the lower bound 2n−2 for the splitting number of the n-cube. It is known that the splitting number of the n-cube is O(2n), thus our result implies that the splitting number of the n-cube is λ(2n).

Automatic Visualization of Two-Dimensional Cellular Complexes

A two-dimensional cellular complex is a partition of a surface into a finite number of elements—faces (open disks), edges (open arcs), and vertices (points). The topology of a cellular complex is the abstract incidence and adjacency relations among its elements. Here we describe a program that, given only the topology of a cellular complex, computes a geometric realization of the same—that is, a specific partition of a specific surface in three-space—guided by various aesthetic and presentational criteria.